Variations on Weyl's theorem

Abstract

In this note we study the property ( w ) , a variant of Weyl's theorem introduced by Rakočević, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property ( w ) holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property ( w ) for T (respectively T * ) coincide whenever T * (respectively T) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property ( w ) .